smile.math.matrix

Class PowerIteration

java.lang.Object

smile.math.matrix.PowerIteration

public class PowerIteration
extends java.lang.Object

The power iteration (also known as power method) is an eigenvalue algorithm that will produce the greatest (in absolute value) eigenvalue and a nonzero vector the corresponding eigenvector.

Constructor Summary

Constructors

Constructor and Description
PowerIteration()

Method Summary

Modifier and Type	Method and Description
static double	eigen(Matrix A, double[] v) Returns the largest eigen pair of matrix with the power iteration under the assumptions A has an eigenvalue that is strictly greater in magnitude than its other eigenvalues and the starting vector has a nonzero component in the direction of an eigenvector associated with the dominant eigenvalue.
static double	eigen(Matrix A, double[] v, double tol) Returns the largest eigen pair of matrix with the power iteration under the assumptions A has an eigenvalue that is strictly greater in magnitude than its other eigenvalues and the starting vector has a nonzero component in the direction of an eigenvector associated with the dominant eigenvalue.
static double	eigen(Matrix A, double[] v, double p, double tol) Returns the largest eigen pair of matrix with the power iteration under the assumptions A has an eigenvalue that is strictly greater in magnitude than its other eigenvalues and the starting vector has a nonzero component in the direction of an eigenvector associated with the dominant eigenvalue.
static double	eigen(Matrix A, double[] v, double p, double tol, int maxIter) Returns the largest eigen pair of matrix with the power iteration under the assumptions A has an eigenvalue that is strictly greater in magnitude than its other eigenvalues and the starting vector has a nonzero component in the direction of an eigenvector associated with the dominant eigenvalue.
static double	eigen(Matrix A, double[] v, double tol, int maxIter) Returns the largest eigen pair of matrix with the power iteration under the assumptions A has an eigenvalue that is strictly greater in magnitude than its other eigenvalues and the starting vector has a nonzero component in the direction of an eigenvector associated with the dominant eigenvalue.

Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Constructor Detail

PowerIteration

public PowerIteration()

Method Detail

eigen

public static double eigen(Matrix A,
                           double[] v)

Returns the largest eigen pair of matrix with the power iteration under the assumptions A has an eigenvalue that is strictly greater in magnitude than its other eigenvalues and the starting vector has a nonzero component in the direction of an eigenvector associated with the dominant eigenvalue.

Parameters:

A - the matrix supporting matrix vector multiplication operation.

v - on input, it is the non-zero initial guess of the eigen vector. On output, it is the eigen vector corresponding largest eigen value.

Returns:

the largest eigen value.

eigen

public static double eigen(Matrix A,
                           double[] v,
                           double tol)

Returns the largest eigen pair of matrix with the power iteration under the assumptions A has an eigenvalue that is strictly greater in magnitude than its other eigenvalues and the starting vector has a nonzero component in the direction of an eigenvector associated with the dominant eigenvalue.

Parameters:

A - the matrix supporting matrix vector multiplication operation.

v - on input, it is the non-zero initial guess of the eigen vector. On output, it is the eigen vector corresponding largest eigen value.

tol - the desired convergence tolerance.

Returns:

the largest eigen value.

eigen

public static double eigen(Matrix A,
                           double[] v,
                           double tol,
                           int maxIter)

Returns the largest eigen pair of matrix with the power iteration under the assumptions A has an eigenvalue that is strictly greater in magnitude than its other eigenvalues and the starting vector has a nonzero component in the direction of an eigenvector associated with the dominant eigenvalue.

Parameters:

A - the matrix supporting matrix vector multiplication operation.

v - on input, it is the non-zero initial guess of the eigen vector. On output, it is the eigen vector corresponding largest eigen value.

tol - the desired convergence tolerance.

maxIter - the maximum number of iterations in case that the algorithm does not converge.

Returns:

the largest eigen value.

eigen

public static double eigen(Matrix A,
                           double[] v,
                           double p,
                           double tol)

Returns the largest eigen pair of matrix with the power iteration under the assumptions A has an eigenvalue that is strictly greater in magnitude than its other eigenvalues and the starting vector has a nonzero component in the direction of an eigenvector associated with the dominant eigenvalue.

Parameters:

A - the matrix supporting matrix vector multiplication operation.

v - on input, it is the non-zero initial guess of the eigen vector. On output, it is the eigen vector corresponding largest eigen value.

p - the origin in the shifting power method. A - pI will be used in the iteration to accelerate the method. p should be such that |(λ₂ - p) / (λ₁ - p)| < |λ₂ / λ₁|, where λ₂ is the second largest eigenvalue in magnitude. If we known the eigenvalue spectrum of A, (λ₂ + λ_n)/2 is the optimal choice of p, where λ_n is the smallest eigenvalue in magnitude. Good estimates of λ₂ are more difficult to compute. However, if μ is an approximation to largest eigenvector, then using any x₀ such that x₀*μ = 0 as the initial vector for a few iterations may yield a reasonable estimate of λ₂.

tol - the desired convergence tolerance.

Returns:

the largest eigen value.

eigen

public static double eigen(Matrix A,
                           double[] v,
                           double p,
                           double tol,
                           int maxIter)

Returns the largest eigen pair of matrix with the power iteration under the assumptions A has an eigenvalue that is strictly greater in magnitude than its other eigenvalues and the starting vector has a nonzero component in the direction of an eigenvector associated with the dominant eigenvalue.

Parameters:

A - the matrix supporting matrix vector multiplication operation.

v - on input, it is the non-zero initial guess of the eigen vector. On output, it is the eigen vector corresponding largest eigen value.

p - the origin in the shifting power method. A - pI will be used in the iteration to accelerate the method. p should be such that |(λ₂ - p) / (λ₁ - p)| < |λ₂ / λ₁|, where λ₂ is the second largest eigenvalue in magnitude. If we known the eigenvalue spectrum of A, (λ₂ + λ_n)/2 is the optimal choice of p, where λ_n is the smallest eigenvalue in magnitude. Good estimates of λ₂ are more difficult to compute. However, if μ is an approximation to largest eigenvector, then using any x₀ such that x₀*μ = 0 as the initial vector for a few iterations may yield a reasonable estimate of λ₂.

tol - the desired convergence tolerance.

maxIter - the maximum number of iterations in case that the algorithm does not converge.

Returns:

the largest eigen value.